Question: Simplify and expand the following expression: $ \dfrac{2}{2a - 14}- \dfrac{4}{3a - 24}- \dfrac{2a}{a^2 - 15a + 56} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{2}{2a - 14} = \dfrac{2}{2(a - 7)}$ We can factor a $3$ out of denominator in the second term: $ \dfrac{4}{3a - 24} = \dfrac{4}{3(a - 8)}$ We can factor the quadratic in the third term: $ \dfrac{2a}{a^2 - 15a + 56} = \dfrac{2a}{(a - 7)(a - 8)}$ Now we have: $ \dfrac{2}{2(a - 7)}- \dfrac{4}{3(a - 8)}- \dfrac{2a}{(a - 7)(a - 8)} $ The least common multiple of the denominators is: $ 6(a - 7)(a - 8)$ In order to get the first term over $6(a - 7)(a - 8)$ , multiply by $\dfrac{3(a - 8)}{3(a - 8)}$ $ \dfrac{2}{2(a - 7)} \times \dfrac{3(a - 8)}{3(a - 8)} = \dfrac{6(a - 8)}{6(a - 7)(a - 8)} $ In order to get the second term over $6(a - 7)(a - 8)$ , multiply by $\dfrac{2(a - 7)}{2(a - 7)}$ $ \dfrac{4}{3(a - 8)} \times \dfrac{2(a - 7)}{2(a - 7)} = \dfrac{8(a - 7)}{6(a - 7)(a - 8)} $ In order to get the third term over $6(a - 7)(a - 8)$ , multiply by $\dfrac{6}{6}$ $ \dfrac{2a}{(a - 7)(a - 8)} \times \dfrac{6}{6} = \dfrac{12a}{6(a - 7)(a - 8)} $ Now we have: $ \dfrac{6(a - 8)}{6(a - 7)(a - 8)} - \dfrac{8(a - 7)}{6(a - 7)(a - 8)} - \dfrac{12a}{6(a - 7)(a - 8)} $ $ = \dfrac{ 6(a - 8) - 8(a - 7) - 12a} {6(a - 7)(a - 8)} $ Expand: $ = \dfrac{6a - 48 - 8a + 56 - 12a}{6a^2 - 90a + 336} $ $ = \dfrac{-14a + 8}{6a^2 - 90a + 336}$ Simplify: $ = \dfrac{-7a + 4}{3a^2 - 45a + 168}$